In an A.P. consisting of 23 terms , the sum of the three terms in the middle is 114 and that of the last three is 204 . Find the sum of first three terms :

Options

A. a. 14
B. b. 42
C. c. 24
D. d. 69

Answer

To solve the problem step by step, we will use the properties of an arithmetic progression (A.P.).

### Step 1: Identify the terms in the A.P.
In an A.P. with 23 terms, the middle term is the 12th term (T12). The three middle terms are T11, T12, and T13.

### Step 2: Set up the equations based on the given information.
1. The sum of the three middle terms (T11 + T12 + T13) is given as 114.
2. The sum of the last three terms (T21 + T22 + T23) is given as 204.

### Step 3: Express the terms in terms of the first term (a) and common difference (d).
- The middle terms can be expressed as:
- T11 = a + 10d
- T12 = a + 11d
- T13 = a + 12d

Thus, we can write:
\[ T11 + T12 + T13 = (a + 10d) + (a + 11d) + (a + 12d) = 3a + 33d = 114 \]
This simplifies to:
\[ 3a + 33d = 114 \] (Equation 1)

- The last three terms can be expressed as:
- T21 = a + 20d
- T22 = a + 21d
- T23 = a + 22d

Thus, we can write:
\[ T21 + T22 + T23 = (a + 20d) + (a + 21d) + (a + 22d) = 3a + 63d = 204 \]
This simplifies to:
\[ 3a + 63d = 204 \] (Equation 2)

### Step 4: Solve the equations simultaneously.
Now we have two equations:
1. \( 3a + 33d = 114 \) (Equation 1)
2. \( 3a + 63d = 204 \) (Equation 2)

Subtract Equation 1 from Equation 2:
\[ (3a + 63d) - (3a + 33d) = 204 - 114 \]
This simplifies to:
\[ 30d = 90 \]
Thus, we find:
\[ d = 3 \]

### Step 5: Substitute the value of d back into one of the equations to find a.
Using Equation 1:
\[ 3a + 33(3) = 114 \]
This simplifies to:
\[ 3a + 99 = 114 \]
Subtracting 99 from both sides gives:
\[ 3a = 15 \]
Thus, we find:
\[ a = 5 \]

### Step 6: Find the first three terms of the A.P.
The first three terms can be calculated as:
- T1 = a = 5
- T2 = a + d = 5 + 3 = 8
- T3 = a + 2d = 5 + 2(3) = 11

### Step 7: Calculate the sum of the first three terms.
The sum of the first three terms is:
\[ T1 + T2 + T3 = 5 + 8 + 11 = 24 \]

### Final Answer:
The sum of the first three terms is **24**.